Generalized Eigenvectors of Isospectral Transformations,Spectral Equivalence and Reconstruction of Original Networks

TL;DR

This paper investigates how generalized eigenvectors behave under isospectral transformations of networks, explores spectral equivalence, and examines the potential for reconstructing original networks from compressed forms.

Contribution

It generalizes the invariance of eigenvectors under isospectral transformations and clarifies the concept of spectral equivalence of networks.

Findings

Eigenvectors are invariant under isospectral transformations

Networks can be reconstructed from their compressed images

Spectral equivalence provides a framework for network comparison

Abstract

Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors.We analyze here what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks.